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Research Article
Received 4 June 2010, Accepted 4 January 2011 Published online 22 March 2011 in Wiley Online Library
(wileyonlinelibrary.com) DOI: 10.1002/sim.4208
Flexible modeling of the effects of
continuous prognostic factors in relative
survival
Amel Mahboubi,a,b Michal Abrahamowicz,a,b∗†Roch Giorgi,c
Christine Binquet,d,e Claire Bonithon-Koppd,e and
Catherine Quantind,e
Relative survival methods permit separating the effects of prognostic factors on disease-related ‘excess
mortality’ from their effects on other-causes ‘natural mortality’, even when individual causes of death are
unknown. As in conventional ‘crude’ survival, accurate assessment of prognostic factors requires testing and
possibly modeling of non-proportional effects and, for continuous covariates, of non-linear relationships with
the hazard. We propose a flexible extension of the additive-hazards relative survival model, in which the
observed all-causes mortality hazard is represented by a sum of disease-related ‘excess’ and natural mortality
hazards. In our flexible model, the three functions representing (i) the baseline hazard for ‘excess’ mortality,
(ii) the time-dependent effects, and (iii) for continuous covariates, non-linear effects, on the logarithm of
this hazard, are all modeled by low-dimension cubic regression splines. Non-parametric likelihood ratio tests
are proposed to test the time-dependent and non-linear effects. The accuracy of the estimated functions is
evaluated in multivariable simulations. To illustrate the new insights offered by the proposed model, we apply
it to re-assess the effects of patient age and of secular trends on disease-related mortality in colon cancer.
Copyright ©2011 John Wiley & Sons, Ltd.
Keywords: time-dependent effects; non-linear effects; regression splines; alternating conditional estimation;
simulations
1. Introduction
Relative survival is an increasingly popular approach for modeling censored survival data in prognostic
studies where the individual causes of death remain unknown or the accuracy of this information
is low such as cancer registry-based studies [1]. To deal with such data limitations, several relative
survival regression models have been proposed to discriminate between the effects of covariates on
disease-related mortality from their effects on other-causes mortality [1--9].
Among alternative relative survival models, the additive hazard model proposed by Estève et al.
[2] is very popular in cancer registry-based studies, e.g. [10, 11]. The model partitions the observed
hazard for all-cause mortality into two additive components: the expected hazard of all-cause mortality
and the ‘excess’ hazard representing disease-related mortality. The former is usually obtained from
vital statistics for the underlying general population, whereas the latter is estimated from the data at
hand [2]. Simulations demonstrated that, in the absence of information about the cause of death of
aDepartment of Epidemiology and Biostatistics, McGill University, Montreal, Quebec, Canada H3A 1A1
bDivision of Clinical Epidemiology, McGill University Health Centre, Montreal, Quebec, Canada H3A 1A1
cLERTIM EA 3283, Faculté de Médecine, Université de la méditerranée, Marseille, F-13005, France
dINSERM, U866, Univ Bourgogne, Dijon, F-21079, France
eDepartment of Biostatistics and Medical Informatics, Dijon University Hospital, France
∗Correspondence to: Michal Abrahamowicz, Division of Clinical Epidemiology, McGill University Health Centre, 687
Pine Ave. West, V-Building, V2.20A, Montreal, Quebec, Canada H3A 1A1.
†E-mail: michal.abrahamowicz@mcgill.ca
Copyright ©2011 John Wiley & Sons, Ltd. Statist. Med. 2011, 30 1351–1365
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A. MAHBOUBI ET AL.
individual subjects, the relative survival model proposed by Estève et al. yields more accurate estimates
of the effects of prognostic factors on disease-related mortality than the conventional Cox’s proportional
hazards (PH) analysis of ‘crude’ all-cause mortality [12].
However, when modeling the covariate effects on the disease-related mortality hazard, the original
Estève et al.’s model retains parametric assumptions of the conventional Cox’s PH [13] model for
‘crude survival’. Specifically, the PH assumption implies that the hazard ratios (HRs) associated with
all covariates remain constant over the entire follow-up period. Furthermore, similar to all conventional
GLM models, each continuous covariate is assumed to have a pre-specified parametric, usually linear,
relationship with the logarithm of the hazard [2]. Yet, several flexible analyses of either relative or crude
survival in various cancers demonstrated that both the PH assumption [5, 14--16], and the log-linearity
assumption for continuous covariates [17, 18], are often inconsistent with the actual effects of several
prognostic factors. Moreover, in the context of crude all-cause mortality analyses, Abrahamowicz and
MacKenzie have recently demonstrated that simultaneous modeling and testing of both PH and log-
linearity hypotheses is essential to avoid biased estimates and incorrect inference about the effects of
continuous prognostic factors [19].
Following these developments in the crude survival methodology, in the last decade, some flexible
extensions of the Estève et al.’s relative survival model have been proposed. Giorgi et al. [5] adapted
the crude survival regression spline modeling of time-dependent effects [20] to relative survival, and
validated the resulting flexible extension of the Estève et al.’s model in simulations. Recently, Remontet
et al. [6] extended that model further, by adding spline-based modeling of a possibly non-linear effect
of a continuous prognostic factor on the log hazard of disease-related mortality. However, while their
model is similar in spirit to the aforementioned crude survival model in [19], an important difference
concerns the assumptions about the relationships between the non-linear and time-dependent effects
of the same continuous covariate. In contrast to [19], where the two effects are assumed to be multi-
plicative, Remontet et al. [6] assume that they are additive. Yet, as in many other areas of statistical
modeling, the distinction between additive versus multiplicative models implies important concep-
tual and computational differences in, respectively, interpretation of the results and model estimation
[21, 22].
The main objectives of this paper are to (i) develop a new flexible multiplicative relative survival
model that extends the Estève et al.’s model [2] and (ii) evaluate the new model in simulations. The
following section starts with a brief review of the Estève et al.’s model [2], and then introduces our
new, flexible multiplicative relative survival model. Then, Section 2.3 describes the iterative alternating
conditional full maximum likelihood estimation procedure, and Section 2.4 discusses hypothesis testing.
Section 3 outlines the design and assumptions of the simulations used to validate our estimates and tests,
and summarizes the results of simulations. Section 4 illustrates an application of our new flexible relative
survival model in the analyses of mortality due to colorectal cancer, based on a cancer registry with
unknown individual causes of death. The paper concludes with a brief discussion of some limitations
of our study, and of the possible directions and challenges for future studies in this field.
2. New flexible model for relative survival
2.1. Parametric additive-hazards relative survival model of Estève et al.
In this paper, we propose a flexible extension of the parametric relative survival regression model
developed by Estève et al. [2]. Their additive-hazards model assumes that the observed hazard for
all-cause mortality, t, at time tafter diagnosis, of an individual with age aat diagnosis and with a
covariate vector z, is the sum of two hazards:
t(t|z,a)=e(t+a|zs)+c(t|z) (1)
The first component, e, represents the expected hazard function for all-causes mortality in the under-
lying general population, stratified on age at diagnosis (a) and a limited subset zsof other covariates,
typically sex and calendar period, and is obtained from relevant mortality statistics [2].
The second component, cin (1), is the hazard function for disease-related mortality, conditional on
the covariate vector z, which is estimated from the data at hand. Estève et al. [2] propose to estimate c
using a parametric model, which is similar to the familiar Cox’s proportional hazards (PH) model but,
in addition, requires estimating the ‘baseline’ log hazard of disease-related mortality, for the ‘reference
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group’ of patients with z=0. The baseline log hazard of disease-related mortality, for patients with z=0
is estimated by dividing the follow-up into Kconsecutive time segments and assuming that hazard is
constant within each segment. Accordingly, the disease-related hazard, conditional on the vector of p
covariates, is expressed as [2]:
c(t|z)=exp p
i=1
izi(t) (2a)
with
(t)=
K
k=1
kIk(t) (2b)
where kis the baseline mortality hazard in the kth time-segment for patients with z=0,Ikis an
indicator function (Ik=1iftk−1t<tk,and Ik=0 otherwise), and i(i=1,...,p) are the log hazard
ratios corresponding to the pcovariates.
Model (2a) imposes the PH assumption, because iare constant, i.e. independent of time t. Moreover,
for quantitative predictors, iis multiplied by an un-transformed predictor value zi, which implies
log-linearity assumption, i.e. a linear relationship between ziand the log hazard.
2.2. New multiplicative model for joint flexible modeling of time-dependent and non-loglinear effects
in relative survival
Our new model respects the assumption of the additivity of the hazards of (i) all-cause and (ii) disease-
related mortality, as postulated by equation (1). The difference between our new model and the Estève
et al.’s model [2] concerns equations (2a) and (2b). In particular, we replace the parametric representation
of both baseline disease-related hazard in (2b), and of covariate effects in (2a), by a more flexible
modeling. First, similar to relative survival models proposed by Giorgi et al. [5] and Remontet et al. [6],
to avoid clinically implausible ‘jumps’ implied by a step function in (2b), we model the baseline hazard
of disease-related mortality, for patients with z=0, corresponding to (t) in (2a), as a smooth function
of the follow-up time. Second, when modeling the effects of continuous covariates on the baseline log
hazard, we simultaneously relax both the PH and the log-linearity assumptions. Specifically, as in the
flexible ‘product model’, for ‘crude’ survival analyses [19], we model the logarithm of the hazard ratio
associated with a value ziof a (continuous) ith covariate, at time t, as a product of two functions:
i(zi)andi(t). Therefore, we model the disease-related hazard, conditional on the covariate vector z,
as
c(t|z)=exp((t)) exp
i
i(zi)i(t)(3)
As in [19], i(zi) in (3) is a possibly non-linear function, i.e. a non-linear transformation of a continuous
covariate zi, the form of which has to be estimated from the data at hand. The estimated non-linear
function i(zi) describes how the log hazard ratio changes with increasing value ziof the ith covariate.
Because in (3), this function does not depend on t,the shape of the non-linear function is assumed to
remain constant over time. However, the impact of the differences in the covariate values does vary
over time because i(zi) is multiplied by the time-dependent function i(t). Higher absolute values of
i(t) indicate that, at a corresponding time t, the differences in the values of zihave stronger impact
on the disease-related hazard.
Similar to [5, 19], all three functions in (3) are estimated by low-dimension cubic regression splines,
i.e. smooth piecewise cubic polynomials, whose pieces join each other at pre-specified argument values
termed ‘knots’ [23]. Using cubic splines ensures the continuity of the estimated function and its first
and second derivatives, while limiting the number of the estimated coefficients enhances the power for
testing non-linear and/or time-dependent effects and reduces the risk of overfitting bias [20]. Therefore,
we use 1 interior knot, i.e. a 2-piece cubic spline, to model the non-linear function i(zi), and 2 interior
knots, corresponding to a 3-piece spline, for both functions of time: the time-dependent function i(t)
and (t), i.e. the baseline log hazard of disease-related mortality, for patients with z=0. Specifically,
we place a single knot at the sample median value of a continuous covariate zifor i(zi), and two
knots at the terciles of the observed distribution of un-censored event times for both i(t)and(t).
To avoid identifiability problems, we need to constrain the function i(zi)tohaveafixedvaluefor
a specific ‘reference’ value of a continuous covariate zi[19]. Accordingly, we impose the restriction
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i[min{zi}]=0, where min{zi}is the minimum value of ziin the sample. This is achieved by constraining
the first coefficient of i(zi) to 0 [19], so that in (4a) only 4 spline coefficients il,l=2,...,5are
estimated for zi. Accordingly, in our model (3):
i(zi)=
5
l=2
i,lBi,l,4(zi) (4a)
i(t)=
6
k=1
i,kBk,4(t) (4b)
(t)=
6
k=1
kBk,4(t) (4c)
where B.,4are cubic, i.e. order 4, B-splines [24], iindexes the covariates, while land kindex splines in
the respective cubic B-splines bases. In particular, for a given covariate zi,lindexes consecutive cubic
B-spline transformation of the covariate values, while kindexes the transformations of the follow-up
time t. Thus, the B-spline basis in (4a) varies depending on the covariate. In contrast, equations (4b)
and (4c) use the same B-spline basis which, in (4b), is identical for all the covariates. However, the
corresponding estimates of the spline coefficients will likely differ, resulting in different shapes of the
(i) estimated i(t) functions for different covariates and (ii) (t).
Equations (4a) and (4b) relate to a continuous covariate, for which both non-linear and time-
dependent effects are modeled. For binary covariates, equation (4a) simplifies to a single parameter
(log hazard ratio) i(zi)=i. Furthermore, once the multivariable flexible model (3) is estimated,
a data analyst may rely on significance tests, described in Section 2.4 below, to possibly simplify
the model. If the linearity hypothesis is not rejected for a given continuous covariate, then its
linear effect on the log hazard may be represented by a single slope parameter i. Similarly,
for either binary or continuous covariates, for which the PH assumption is not rejected, the
corresponding i(t) in equation (4b) may be replaced by i, i.e. constant-over-time log hazard
ratio.
Notice that, in some applications, data analysts may choose different numbers of interior knots
for some or all of the three functions in (4a)–(4c). However, to obtain stable estimates and accurate
inference, it is important to ensure that the ratio of the number of deaths expected to arise from
disease-related mortality to the total number of model’s degrees-of-freedom exceeds 10 [19].
2.3. Alternating conditional estimation
Estimation of model (3) poses another un-identifiability problem because the two functions of the same
covariate i(zi)andi(t) ‘share the scale’ [19]. Indeed, for any constant i=0, equation (3) can be
rewritten as
c(t|z)=exp((t)) exp
i
ii(zi)i(t)
i
Abrahamowicz and MacKenzie [19] circumvented this problem by aprioriconstraining the range of
the estimated values of i(zi) to 1. This ‘scale constraint’ implies that a value of i(t), at a given
time t, represents the adjusted difference in log hazards between the values of zicorresponding to,
respectively, the highest and the lowest risks [19]. However, in the joint estimation of the crude
survival model, implementation of the scale constraint required a complex trial-and-error procedure
and caused occasional non-convergence problems [19]. Below, we propose a simpler estimation algo-
rithm, which avoids the non-identifiability problems and directly restricts the estimated range of i(zi)
values to 1.
Specifically, we propose to replace simultaneous estimation of all 3 functions in (4a)–(4c) by an
iterative alternating conditional algorithm. Alternating conditional algorithms are especially useful for
estimation of complex models, with specific constraints imposed on only a subset of estimable coeffi-
cients. Such subset-specific constraints make it difficult to simultaneously estimate all the coefficients of
interest [25, 26]. The alternating conditional algorithm avoids such difficulties by splitting the estimation
into consecutive steps, each of which involves estimation of only a subset of all coefficients of interest.
The algorithm iterates between the steps, and in each step only a corresponding subset of coefficients
are estimated, conditional on the fixed values of all other coefficients, set to their most recent estimates.
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Table I. Performance of the alternating conditional algorithm applied in the analysis of colon cancer
mortality (see Section 4).
(L) Negative L for each L for each
Main iteration Step Estimated effects∗log likelihood step†iteration‡
0—649.5135 — —
11.1 (t) with no covariates 605.7315 43.7820
11.2 (z)|ˆ
from 1.1 584.7770 20.9545
11.3 (t)|ˆ
(1.1),ˆ
(1.2) 580.1794 4.597 25.8997
22.1 (t)|ˆ
(1.2),ˆ
(1.3) 579.8318 0.3476
22.2 (z)|ˆ
(2.1),ˆ
(1.3) 578.2806 1.5512
22.3 (t)|ˆ
(2.1),ˆ
(2.2) 577.5504 0.3024 2.629
... ... ... ... ... ...
29 29.3 (t)|ˆ
(29.1),ˆ
(29.2) 574.5055
30 30.1 (t)|ˆ
(29.2),ˆ
(29.3) 574.5049 0.0006
30 30.2 (z)|ˆ
(30.1),ˆ
(29.3) 574.5048 0.0001
30 30.3 (t)|ˆ
(30.1),ˆ
(30.2) 574.5044 0.0004 0.0011
31 31.1 (t)|ˆ
(30.2),ˆ
(30.3) 574.5039 0.0005
31 31.2 (z)|ˆ
(31.1),ˆ
(30.3) 574.5038 0.0001
31 31.3 (t)|ˆ
(31.1),ˆ
(31.2) 574.5035 0.0003 0.0009
∗Effect estimated at a given step, conditional on effects show after ‘|’, e.g. in step 2.2, (z)|ˆ
(2.1), ˆ
(1.3)
means that (z) is estimated conditional on ˆ
(t) from step 2.1 and ˆ
(t) from step 1.3.
†Reduction in negative log likelihood (L) observed at a given step, relative to the previous step.
‡Reduction in negative log likelihood (L) observed across the 3 steps of a given iteration, relative to the
previous iteration.
The iterations end once a pre-specified convergence criterion is met. We used the alternating conditional
algorithms in our earlier work on estimation of complex regression spline-based models, where various
constraints had to be imposed on only a subset of estimated coefficients [25, 26]. Simulation results
reported in both papers indicated a satisfactory performance of the alternating conditional algorithm
[25, 26].
Accordingly, we estimate model (3) through a 3-step iterative alternating conditional full maximum
likelihood estimation (MLE) algorithm. Each step estimates the coefficients of one of the three func-
tions in (4a)–(4c), conditional on the fixed values of all the coefficients of the two other functions,
corresponding to their most recent estimates. Specifically, the 3 steps involve flexible estimation of,
respectively:
(1) baseline log hazard function (t) in (4c), conditional on the estimates, obtained in the previous
iteration, of: (i) non-linear function i(zi) in (4a) and (ii) time-dependent functions i(t) in (4b);
(2) non-linear function i(zi), conditional on the estimates of (i) (t) from step (1) of the same
iteration and (ii) i(t) from the previous iteration;
(3) time-dependent functions, conditional on the estimates of (i) (t) and (ii) i(zi) from, respectively,
steps (1) and (2) of the same iteration.
Iterations stop once the reduction in the negative log likelihood, relative to the previous iteration, is
less than 0.001. Table I illustrates the performance of the alternating conditional estimation algorithm
in real-life analyses and shows the improvements in the model’s log likelihood achieved at each of the
3 steps of each iteration.
Notice that the above algorithm avoids the non-identifiability problems because the estimation of the
two functions, i(zi)andi(t), that are multiplied by each other in equation (3), is done separately in,
respectively, steps (2) and (3) of each iteration. The aforementioned ‘scale constraint’ is imposed by
multiplying all coefficients in (4a), estimated in step (2) of each iteration, by a constant Di, such that
the range of the estimated values of i(zi) is constrained to 1.
The details of the algorithm, as well as the initial values selection, are provided in Section 2 of the
online material‡.
‡Supporting information may be found in the online version of this article.
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To implement the above algorithm, we wrote a customized program in R. To implement the additivity
of the natural and disease-related hazards in (1), we adapted the RSurv software, originally developed
by Giorgi et al. [27]. At each step, our program employs the R function NLM to minimize the full
negative log likelihood of model (3) [28]. The R program that implements our model is available on
request from the authors.
2.4. Testing non-linear and time-dependent effects
Our new model (3) permits estimating both non-linear and time-dependent functions for the rela-
tionship between a continuous covariate and the log hazard of disease-related mortality. However,
the end user may be reluctant to accept such complex functions unless they significantly improve
the fit to data, relative to simpler alternative functions [15]. For example, if, for a covariate zp,
the time-dependent effect is statistically non-significant but there is a significant violation of the
log-linearity hypothesis, the effect of zpmay be best represented by a constant-over-time non-
loglinear effect p(zp) [19]. Therefore, it is important to accurately test both the PH and log-linearity
hypotheses.
Polynomial regression splines in equations (4a) and (4b) permit using conventional inferential tools
of large-sample maximum likelihood estimation [29]. In particular, the cubic spline used to estimate the
non-linear effect i(zi) in (4a) includes a linear function izias its special case, while the time-dependent
estimator i(t) in (4b) includes the constant i[20]. Because each of the simpler, conventional models
is nested within the corresponding cubic spline model, the differences between their deviances may be
tested using non-parametric likelihood ratio tests (LRTs) [6, 19]. For example, to test the time-dependent
effect of zi, while adjusting for its non-loglinear effect, we first fit the full product model (3), which
estimates both effects of zi, while adjusting for the required effects of other covariates. Then, we fit a
reduced model, with the same covariate effects except that, while modeling the effect of zi, we replace
the time-dependent estimator i(t) with the constant i, which is absorbed in the non-linear fixed-over-
time function i(zi). We then compare the deviances of the two models. Under the null hypothesis of
the PH effect for zi, the resulting LRT has a chi-square distribution with 6 degrees-of-freedom (df),
because it compares a (i) 10-df ‘product model’ i(t)i(zi) against (ii) the 4-df model i(zi). Similar
LRT tests can be constructed to test the log-linearity, or to compare the flexible product model (3) with
the conventional log-linear PH model izi[19].
Boxes in Figure 1 identify all relevant models that may be considered for a continuous covariate
(here: age in the colon cancer example analyzed in Section 4), with all models shown above a given
model being nested within it. Arrows in Figure 1 identify LRTs for testing various effects, together
with the corresponding dfs.
Non-parametric LRTs tend to yield slightly inflated type I error rates, reflecting the analytical
difficulties in quantifying exact dfs [6, 19, 30]. Therefore, we suggest using a slightly conservative
nominal =0.04 for testing both the PH and loglinearity hypotheses.
Figure 1. Alternative (nested) models for age at cancer diagnosis (see Section 4) and LR tests of ‘non-parametric’
effects. Each model is nested within all models shown below it. Each arrow corresponds to an LR test that
compares the deviances of the two respective (nested) models. (LRT =likelihood ratio test, PH=proportional
hazards, LL=log-linearity, df =degrees of freedom).
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3. Simulations design and results
3.1. Outline of data generation procedures and underlying assumptions
We performed simulations to assess the accuracy of the estimation of both the time-dependent function
(t) and the non-linear function (z) of a continuous covariate. Specifically, we simulated a 5-year
prognostic cohort study of mortality in N=2000 incident colon cancer cases, for whom the individual
causes of death were assumed unknown. The distributions of the prognostic factors and most of their
effects on mortality were based on real-life data from a French colon cancer registry [15].
Section 3 of the online material describes in detail the data generating procedures and the underlying
assumptions. Briefly, data generation involved the following steps:
(1) Generation of Nbaseline covariate vectors, which included three categorical covariates: sex,
cancer location (right vs left colon), and cancer stage (stage II vs III); and a continuous covariate
of primary interest, age at diagnosis, distribution of which varied by sex.
(2) Generation of Nexpected times to cancer-related death, from the marginal distribution with a
gradually decreasing hazard, independent of any covariates.
(3) Matching each of the Ncovariate vectors generated in step (1) with one of the Ntimes to cancer-
related death from step (2), using the ‘permutational algorithm’ we have recently developed and
validated [31, 32]. Matching was performed so that the probability of a cancer-related death at
time tbeing assigned to a subject with a covariate vector z, was proportional to the corresponding
hazard ratio HR(t|z) calculated assuming the following ‘true’ model for hazard of cancer-related
mortality
HR(t|z)=exp[ln(1.25)sex+ln(1.2)location+ln(3)stage+age(age)age (t)] (5)
where age(age) and age (t) represent, respectively, the non-linear and the time-dependent functions
for age at diagnosis. Three simulated scenarios were considered, each with a different combination
of the two functions. (The ‘true’ functions are shown with white circles in respective panels in
Figure 2 and the corresponding equations are given in Section 3 of Supplementary Material.)
(4) Generation of Nexpected times to ‘natural’ other-cause death, conditional on subject’s age and
sex generated in step (1), but otherwise independent of time to cancer-related death. The times to
‘natural’ death were generated based on the life tables for the Côte d’Or administrative region in
France, stratified by sex and age.
(5) Imposing an ‘administrative censoring’, at 5 years of follow-up, of all subjects who remained
‘alive’ until that time.
(6) Determining the observed follow-up time, and survival status, for a given subject, based on the
minimum of: (i) expected time to cancer-related death from step (3), (ii) expected time to ‘natural’
death from step (4), and (iii) administrative censoring at 5 years.
Notice that, because causes of death are assumed unknown, in the final analysis data set, no distinction
was made between cancer-related versus ‘natural’ deaths.
3.2. Results of simulations
We analyzed each simulated sample with our new flexible multivariable relative survival model (3).
The expected hazard mortality, ein (1), was quantified based on published age-, gender-, and year of
death-specific mortality rates in the general population of the French region of Côte d’Or, which were
also used to simulate times to ‘natural’ deaths in step (4) of Section 3.1.
The multivariable model used in all analyses incorporated cubic spline modeling of the baseline log
hazard function, (t) in (4c), and of both non-linear, (age) in (4a), and time-dependent, age(t) in (4b),
functions of age at diagnosis. The estimated effects of the three binary prognostic factors were apriori
constrained to be constant over time, consistently with the PH assumption.
3.2.1. Recovery of the true time-dependent and non-linear functions. The main goal of the simulations
was to assess whether our model (3) accurately recovered the true shapes of the time-dependent and
non-linear functions for the effect of age on the log hazard of cancer-specific mortality. Figure 2
summarizes selected results for the three simulated scenarios, which differed in the assumptions about
the ‘true’ shape of (age) and/or age(t). First, to assess the performance of our method in individual
samples, Figures 2(a) and (b) compare the true functions (white circles), for the first scenario, with 100
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Figure 2. Results of the three simulated scenario for a continuous covariate (‘age’). Panels (a) and (b) show
100 estimated curves and the true curve (white circles) for (t)and(age) in scenario 1. Panels (c) and (d)
show mean estimates (bold curve) with the 5th and the 95th percentiles of the pointwise distributions of the
estimated values (dashed curves), and the true curve (white circles) for (t)and(age). Panels (e) and (f) show
estimates (bold curve) with 90 per cent bootstrap-based pointwise confidence bands (dashed curves) and the
true curve (white circle) for (t)and(age) for a random sample simulated for scenario 1. Panel (g) shows 100
estimated curves and the true curve (white circles) for (t) for scenario 2 and panel (f) shows 100 estimated
curves and the true curve (white circles) for (age) for scenario 3.
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sample-specific estimates (solid curves). Figure 2(a) shows a satisfactory performance of cubic spline
estimates of time-dependent function age(t), except for some over-fit bias in the tails of the time axis.
This reflects the well-known instability of regression splines in their tails [5, 6, 19, 20, 23]. Similarly,
Figure 2(b) shows that all estimates of (age) recover correctly the true U-shaped curve.
Figures 2(c) and (d) show that the mean of the 100 estimates (solid curve) of, respectively, age(t)
and (age), are quite close to the corresponding true functions for the first scenario, suggesting that
the estimates are practically free of systematic bias. To assess the local precision of the estimates, the
dashed curves in Figures 2(c) and (d) trace the 5th and the 95th percentiles of the pointwise distributions
of the 100 estimates, at a specific value of time tor age, which allow assessing the local precision of
the estimates. As expected, the empirical variance of age(t) estimates increases in the right tail of the
survival time distribution (Figure 2(c)).
Figures 2(e) and (f) illustrate the performance of the 90 per cent pointwise bootstrap-based confidence
bands, estimated from 100 resamples for, respectively, (t)and(z), in a single random sample from the
first simulated scenario. For both functions, the 90 per cent pointwise bootstrap-based confidence bands
(dashed curves) always include the true function. Figure 2(e) indicates that the bootstrap pointwise
confidence bands correctly reflect the inflated variance of the estimates of the time-dependent function
(t) in the upper tail of the follow-up time distribution, seen in Figure 2(a).
Figure 2(g) compares the true time-dependent function (white circles) for the second scenario with
100 sample-specific estimates (solid curves) of age(t), while Figure 2(h) compares the true non-linear
function (white circles) for the third scenario with 100 sample-specific estimates (solid curves) of
(age). In both graphs, most of the estimates trace closely the corresponding true functions. This
provides further evidence that our model (3) yields reasonably accurate estimates of complex effects
of continuous predictors on the hazard of disease-related mortality, even in the absence of information
about the cause of death.
3.2.2. Sensitivity analyses: assessing the convergence properties of the algorithm. In sensitivity anal-
yses, for scenario 1, we reversed the order of steps 2 and 3 of our alternating conditional algorithm (see
the end of Section 2.3), i.e. in each main iteration, we estimated age(t)before estimating (age). In
all 100 samples, the final values of the (full) log likelihood obtained with the original and the ‘reverse’
versions of the algorithm were uniformly very similar. Indeed, the median absolute difference was 0.007
(relative difference of only 0.0002 per cent), with the largest difference of 0.16 (<0.003 per cent).
Figure 3 provides further insight into the robustness of the estimates, with respect to the order of
the 2 steps. Different panels of Figure 3 compare the corresponding estimates of either (z)or(t)
for two simulated samples, in which the difference in the log likelihood yielded by the two versions
of the algorithm was the smallest (Figures 3(a) and (b)) and the largest (i.e. the ‘worse’) (Figures 3(c)
and (d)) among all 100 samples for scenario 1. In all graphs, the shapes of the two corresponding
estimates are identical, and even in the ‘worse’ sample the differences between the estimated values are
practically negligible (Figures 3(c) and (d)). Thus, comparisons of both (i) the final log likelihood, and
(ii) the estimated functions suggest that, regardless of the order of the 2 steps, the algorithm converges
to practically the same final solution.
3.2.3. Hypothesis testing. To assess the empirical type I error rates of the proposed non-parametric
LR tests (see Section 2.4), we simulated data similar to the first scenario, but assumed that respective
null hypotheses of either (i) PH or (ii) log-linearity were true. Specifically, (i) when testing the PH
hypothesis, we assumed that the non-linear function (age) for age (described by white circles in
Figure 2(b)) remained constant over time. Conversely, (ii) when testing the log-linearity, we assumed
the relationship between age and log hazard was linear but its strength changed over time as described
by (t) (white circles in Figure 2(a)). For each of the 100 samples, simulated under these assumptions,
we then compared the deviance of the full flexible product model in (3) versus the ‘reduced model’,
which imposed the constraints of, respectively (i) PH ((t)=), and (ii) log-linearity ((age)=age).
For both tests, we used the corresponding non-parametric LR tests, with dfs shown in Figure 1, at the
corrected =0.04 (see Section 2.4). The 6-df LR test of the PH hypothesis rejected the true H0 of
the PH effect of age in only 3 of the 100 samples, yielding empirical type I error of 0.03 (95 per cent
CI: 0–0.06). The 4-df LR test of the log-linearity hypothesis yielded type I error of 0.1 (95 per cent
CI: 0.04–0.16).
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Figure 3. Estimates or (t) (panels (a) and (c)) and (age) (panels (b) and (d)) obtained with two versions of
the algorithm in simulated samples where the differences between the corresponding log likelihood values are
the smallest (panels (a) and (b)) and the largest (i.e. the ‘worse’) (panels (c) and (d)).
4. Real-life application: re-assessment of the role of prognostic factors for
mortality in colon cancer
4.1. Data source
To illustrate the application of our method in the real-life context, we re-assess the role of selected
prognostic factors for mortality in colon cancer. The data obtained from the Digestive Cancers Registry
of Burgundy (France) included 813 residents of the Côte d’Or administrative region, diagnosed with
stage I colon cancer [33], between 1976 and 2000, and resected for cure. Time 0 was the date of
the cancer diagnosis. The cause of death was not recorded. The follow-up of individual patients was
restricted to the first five years after diagnosis, when all patients still alive were censored. There were
219 deaths of any cause during the first five years since diagnosis (73.1 per cent censoring rate).
In all analyses, we included two continuous available prognostic factors: age at diagnosis (mean=
69.3 years, median =71, range: 31–95), and calendar period of diagnosis (range: 1976–2000), and two
binary variables: gender and tumour location (right vs left colon). Period of diagnosis was included to
investigate whether survival of colon cancer patients did change between 1976 and 2000 and, if so, to
assess the pattern of such changes.
Table I of the online material summarizes the distributions of the prognostic factors and shows the
corresponding number of deaths of any cause during the 5 years after diagnosis.
4.2. Flexible relative survival analyses
In flexible multivariable analyses that used our ‘product model’ (3) for relative survival, for both age
and calendar period, we modeled both non-linear and time-dependent functions defined, respectively,
in equations (4a) and (4b). In contrast, the estimated effects of gender and cancer location were a
priori constrained to the PH assumption, consistent with previous analyses [5]. Finally, the baseline
log hazard (t) was modeled by the cubic spline expansion in (4c).
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Table I illustrates the performance of the iterative alternating conditional estimation algorithm.
Consecutive rows show how the (full) negative loglikelihood changes across the 3 steps of each iteration
and the consecutive ‘main iterations’. The iterations stopped when the improvement in the loglikelihood,
relative to the previous main iteration, was smaller than 0.001 (see Section 2.3). This pre-specified
convergence criterion was reached after 31 main iterations. As expected, the improvements in the
loglikelihood were very important in the first main iteration and relatively minor thereafter. Yet, in the
30 later iterations the loglikelihood improved by almost 6.0, and, thus, the deviance by 12.0.
In sensitivity analyses, the two versions of the algorithm (see Section 3.2.2), yielded very similar
final loglikelihoods: 574.5035 for the original algorithm vs 574.4994 for the ‘reverse’ order of steps
2 and 3. Furthermore, as shown in Figure 2 of the online material, all the estimated functions were
practically identical.
We first focused on LR testing of the non-parametric effects of the two continuous covariates (see
Section 2.4). Figure 1 summarizes the results of different tests for age at diagnosis. Comparison of
the results of different tests demonstrates the importance of accounting simultaneously for possible
violations of both (i) the PH and (ii) the log-linearity hypotheses. Indeed, the p-values in the upper
part of Figure 1 show that (i) the test of the PH, under the log-linearity constraint (p=0.08) and
(ii) the test of log-linearity under the PH constraint ( p=0.21), both yielded non-significant results.
In contrast, the lower part of Figure 1 indicates that both tests yielded significant results (p=0.017
for PH and p=0.038 for log-linearity) when both constraints were simultaneously relaxed. Thus, by
accounting simultaneously for time-dependent and non-linear effects of age, our product model (3)
allows us to detect the statistical significance of both effects. Indeed, the 9-df ‘global’ non-parametric
test ( p=0.02 at the centre of Figure 1) shows that the flexible product model fits significantly better
than the parametric model that imposes both the PH and log-linearity constraints. In contrast, for year
of diagnosis, neither the PH ( p=0.84) nor the log-linearity ( p=0.86) hypotheses were rejected, and
the estimates indicated a steady reduction in the cancer-related mortality between 1976 and 2000 (data
not shown).
The top two panels of Figure 4 describe the adjusted effect of age at diagnosis on the log hazard
of colon cancer-related mortality. The vertical bars correspond to 90 per cent pointwise confidence
bands (CIs), based on 200 bootstrap resamples, and reflect the uncertainty about estimated (age) and
(t) values at selected values of, respectively, age and follow-up time. The time-dependent function
in Figure 4(a) indicates that age at diagnosis has a very strong impact on very early mortality, mostly
due to post-surgery complications. The fact that even the lower bounds of the corresponding pointwise
CIs are above 0 underscores the statistical significance of this early impact. Age also plays a role in
the second year after diagnosis, when the pointwise CIs also exclude 0, even if their width indicates
a considerable lack of precision in the estimated (t) (Figure 4(a)). This numerical instability of the
estimates for t>10 months reflects the combination of (i) the limited total number of deaths (219)
observed in our study, (ii) an important decrease in mortality after the first few months since diagnosis,
and (iii) the fact that many of these later deaths are likely due to the natural ‘background’ mortality
rather than to colon cancer. Later, the impact of age at diagnosis seems to decrease to become practically
nil at >25 months since diagnosis, although the over-fit bias and the very wide CIs make it difficult to
interpret the right tail of the curve.
Figure 4(b) shows how the log hazard changes with increasing age at diagnosis, relative to the
minimum age of 31 years. The non-monotone J-shaped estimate suggests that cancer-related mortality
increases for both very young and older subjects, with the lowest risk for those diagnosed at 40–50 years.
Yet, due to (i) relatively low number of deaths and (ii) low proportion of patients diagnosed with colon
cancer at age below 55 years, the pointwise confidence bands for log HR at ages 40–55, relative to
31 years, include 0. The test of the non-linear effect of age however indicates that the non-monotone
estimate in Figure 4(b) fits the data significantly better ( p=0.038) than the conventional linear estimate
implying that hazard of cancer-related mortality increases monotonically with increasing age. It should
be noted, however, that the confidence bands for the relative survival estimates (Figures 4(a) and (b))
are systematically much wider than the corresponding bands for the crude survival estimates (Figures
4(c) and (d)). This increased variance of the relative survival estimates has also been found in previous
simulations that compared Estève et al.’s relative survival model [2] with Cox’s crude survival model
[12]. This variance inflation is partly due to the fact that while crude survival uses information about
all observed deaths, only deaths expected to reflect disease-related survival contribute to the estimation
of covariate effects in relative survival. To assess the strength of the age effect, at different times tafter
diagnosis, the values in Figure 4(b) have to be multiplied by the corresponding values of (t) from
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Figure 4. Results for the continuous covariate ‘age at diagnosis’ (in the cohort of 813 stage I colon cancer
patients in the real-life application) obtained with our new multiplicative relative survival model (upper part)
and the crude survival model developed in [19] (lower part) for: Time-dependent function age(t) (panels (a)
and (c)) and non-loglinear function age(age) (panels (b) and (d)). Vertical bars correspond to the 90 per cent
pointwise confidence bands, based on 200 bootstrap resamples, for the estimated values of age(t)orage (age)
at selected values of, respectively, follow-up time tor age.
Figure 4(a) (see Section 2.2). For example, at t=2 months, when (2)=3.6, patients diagnosed at age
70 ((70)=0.33) are estimated to have the HR of exp{3.6∗[0.33−(−0.43)]}=15.4 relative to 45 olds
((45)=−0.43). However, at 2 years after diagnosis the corresponding HR decreases to 4.1, because
(24)=1.85.
The lower half of Figure 4 shows the corresponding functions for the effect of age on all-cause
mortality, estimated with the flexible product model for crude survival [19] that adjusted for the
same covariates. (Notice that the pointwise CIs in Figure 4(d) are very narrow at ages close to the
mean age of about 70 years, because the non-loglinear function (z) in the crude survival model in
[19] is constrained to equal 0 at the ‘reference value’ corresponding to the mean covariate value.)
The time-dependent crude survival estimate in Figure 4(c) does not exhibit a high peak right after
diagnosis, seen in Figure 4(a), and there is no evidence of changes over time in the effect of age
(p=0.19 for test of PH). Furthermore, Figure 4(d) suggests that, with increasing age, all-cause mortality
increases monotonically, and approximately linearly ( p=0.07 for log-linearity test), in contrast to a
non-monotone J-shaped relative survival estimate in Figure 4(b). Indeed, some differences between the
relative and crude survival estimates were expected because the latter approach ignores time-varying
‘natural’ mortality, which may account for a large proportion of observed deaths, especially in studies
with longer follow-up and/or more benign disease, such as stage I colon cancer [1, 2, 12].
5. Discussion
We have developed a new flexible model for relative survival analyses of the effects of continuous
prognostic factors on disease-related mortality. Our model generalizes the additive hazards model of
Estève et al. [2] and allows for simultaneous flexible modeling and testing of both non-linear and
time-dependent functions. Similar to the ‘crude survival’ model of Abrahamowicz and MacKenzie
[19], we assume the multiplicative effects of the two functions on the log hazard. Simulation results
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help in evaluating the accuracy of the estimates of both functions and tests of the corresponding
hypotheses.
The colon cancer application illustrates the new insights that the proposed model may offer in real-
life prognostic studies. First, it confirms the importance of simultaneous modeling and testing of both
non-linear and time-dependent functions [19]. For age at cancer diagnosis, when (i) the log-linearity
hypothesis was tested while imposing the (incorrect) PH constraint and, in a separate model, (ii) the
PH was tested assuming (incorrectly) linearity, neither hypothesis was rejected. In contrast, both (i)
the non-linear and (ii) the time-dependent effects became significant in the flexible product model (3),
which simultaneously accounted for both effects. Second, the comparison with a similar flexible product
model for crude survival [19] demonstrated that accounting for natural mortality may substantially alter
the conclusions regarding prognostic factors’ effects. Indeed, crude survival analyses are expected to
yield estimates different from relative survival models [1, 2, 12], in which the likelihood is modified to
account for the time-varying ‘natural’ background mortality (see Section 1 of Supplementary Material).
For example, in crude analyses, hazard increased monotonically with increasing age, in contrast to a
non-monotone J-shaped relative survival estimate. Thus, the impact of older age on all-cause mortality
‘masked’ an important increase in cancer-related mortality for very young colon cancer patients, who
may have a particularly aggressive cancer. Furthermore, in contrast to relative survival estimate, the
time-dependent crude survival estimate did not indicate the high impact of age on mortality right after
diagnosis, which likely reflects the vulnerability of older patients to post-surgical complications. These
important differences between relative and crude survival analyses are not surprising given that as many
as 127 (58 per cent) among the 219 deaths observed in our study could be expected due to ‘natural’
mortality (data not shown).
Some limitations of our method require comments. Low-dimension regression splines with fixed
knots have limited flexibility so that the estimates may only approximate the actual impact of increasing
the covariate value and/or the actual changes over time in the hazard ratios. Indeed, Figure 2 shows
such ‘local bias’ in the estimates from individual samples, even if the mean estimates are relatively un-
biased. However, such approximate estimates are typically sufficient both for clinical prognosis and for
assessing the role of particular prognostic factors [15]. Moreover, the model parsimony increases power
for hypothesis testing [6, 19, 30] and improves the bias/variance trade-off, especially in moderately
sized studies, such as our colon cancer application. In larger studies, where the continuous covari-
ates may have more complex effects, the user may consider alternative models with say, 1–3 interior
knots, and then select the best-fitting estimate through goodness-of-fit criteria such as AIC or BIC.
Our method allows for an arbitrary number of knots for each of the three functions in (4a)–(4c).
However, if the ‘optimal’ number(s) of knots for the final model are selected a posteriori, then this
has to be accounted for at the hypothesis testing stage, to avoid inflated type I error [6, 20]. Finally,
while alternative smoothers such as penalized smoothing splines reduce ‘local bias’ [30], the quasi-
parametric nature of regression splines facilitates hypothesis testing [6, 19, 29]. Indeed, end users may
be reluctant to accept more complex statistical models unless they significantly improve the prediction
of outcomes [15, 18]. In our simulations, the proposed non-parametric LR tests were rather conser-
vative for testing the PH hypothesis, but the opposite was true for testing log-linearity. Interestingly,
linearity tests based on other flexible models have also inflated type I error, likely due to analytical
difficulties in quantifying the exact degrees-of-freedom [19, 30]. Further simulations are necessary to
estimate the type I error rates more precisely, and to assess the power of both tests, under different
assumptions.
In our multiplicative relative survival model (3), the shape of the non-linear function i(zi) does not
change over time. Yet, for example, increased risk associated with cancer diagnosis at a very young
age [17, 18] may not apply to early post-surgical mortality, so that the J-shaped estimate in Figure 4(b)
may represent only the impact of age on later mortality. Thus, future research should develop efficient
tests of the shape-invariance assumption. This issue should be investigated in the broader context of
developing reliable and practically useful criteria to assess goodness-of-fit of relative survival models,
possibly by adapting the methods recently proposed by Stare et al. [34], Sasieni [8] and Cortese and
Scheike [9]. Furthermore, time-varying covariates can be incorporated in our model (3) by using their
relationship with time-dependent effects [20]. Finally, because model (3) is non-linear in its parameters,
it is difficult to obtain analytical confidence bands around the two estimated functions or the 3D surface,
representing their product [19]. In our simulations, bootstrap-based pointwise confidence bands reflected
reasonably well the empirical pointwise variance of the estimates. Further research should assess their
accuracy in a larger range of scenarios.
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Furthermore, our 3-step iterative alternating conditional estimation algorithm is not theoretically
proven to converge to the actual global maximum of the joint likelihood of model (3). However, in
sensitivity analyses, where we reversed the order in which (z)and(t) were estimated, the median
difference between the log likelihoods obtained with (i) original versus (ii) ‘reverse’order in which
(z)and(t) were estimated was only 0.007 (relative difference of 0.0002 per cent), with the largest
difference of 0.16 (<0.003 per cent). Equally important, the estimated functions were virtually the
same, even in the sample with the largest difference. This suggests that the algorithm converges to
practically an identical final solution, regardless of the order of its steps.
We add another method to an already growing toolbox for flexible modeling of relative survival
[5, 6], which includes methods for multivariable modeling of time-varying effects and non-linear trans-
formations of several continuous covariates [7--9]. Our model (3) can be considered a combination of
the spline-based time-dependent relative survival model developed by Giorgi et al. [5] and the ‘crude’
survival model for joint modeling, proposed by Abrahamowicz and MacKenzie [19]. Our work on
simultaneous modeling in crude [19] and relative survival [35] has proceeded in parallel to a develop-
ment of a similarly flexible relative survival model by Remontet et al. [6]. These authors also extend
Estève et al.’s model [2] to include regression spline-based estimation of both disease-related mortality
hazard and covariate effects. Yet, the two models differ substantially in the way time-dependent and
non-linear functions of the same covariate are combined. Remontet et al. model the effect of age on
the logarithm of disease-related hazard by two additive terms: g(age) represents non-linear relationship
between age and log hazard at time 0, while h(t)∗age captures the time-dependent interaction between
age and a flexible follow-up time transformation h(t) [6]. However, the time-dependent interaction
h(t)∗age is limited to a linear function of age. Thus, while the sum of two functions remains non-linear,
the shape of the resulting relationship between age and log hazard varies over time. Indeed, the stronger
the interaction effect h(t), the more linear the estimated effect of age will appear at the corresponding
time t, as the linear component of the sum g(age) +h(t)∗age outweighs the non-linear term g(age). In
fact, this ‘linearization’ may be seen in Figure 3 of Remontet et al.’s article, where the strongest effect
of age seems nearly linear [6].
In conclusion, further simulations and empirical studies are necessary to systematically compare
the flexible additive relative survival model proposed by Remontet et al. [6] with our multiplicative
model (3). Most importantly, both models represent further, potentially complementary, refinements
of the relative survival methodology. Such developments may enhance our understanding of disease
progression and mortality, and ultimately improve the clinical prognosis and management of various
cancers.
Acknowledgements
This research was supported by the Canadian Institutes for Health Research (CIHR) grant # MOP-81275 (PI
M. Abrahamowicz). A. Mahboubi was initially supported by the Regional Council of Burgundy and INSERM,
M. Abrahamowicz is a James McGill Professor at McGill University.
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